University differential equations test to download as PDF

This University differential equations test lasts 60 minutes and is marked out of 20. It covers the whole chapter for pupils aged 18-19, and the full answers are on a separate page.

⏱️ Duration : 60 min
🎯 Total : 20 marks
❓ 5 questions

Mark/20

Question 1 – A separable equation
Solve \(\dfrac{dy}{dx}=\dfrac xy\) with \(y(0)=2\), giving \(y\) explicitly for \(y>0\).
Question 2 – A first-order linear equation
Solve \(y'+y=2e^x\) with \(y(0)=2\).
Question 3 – A second-order equation
Solve \(y''-4y'+4y=0\) with \(y(0)=1\) and \(y'(0)=3\).
Question 4 – Problem: cooling metal
A metal bar at \(200^\circ\)C is placed in a workshop at \(20^\circ\)C. Its temperature satisfies \(\dfrac{dT}{dt}=-k(T-20)\). After 10 minutes the bar is at \(110^\circ\)C.
1. Show that \(T=20+180e^{-kt}\) and find \(k\). (3 marks)
2. Find when the bar reaches \(50^\circ\)C. (1 mark)
Question 5 – Forced oscillation
Find the general solution of \(y''+y'-2y=6e^{2x}\).

How to use this University differential equations test

First, revise with the differential equations lesson and the worksheet. Then sit the test in one go, with a timer set to 60 minutes. After that, mark your work with the differential equations test answers. If you lose marks on a question, redo it the next day without help.

Finally, you can download this University differential equations test as a PDF and print it like a real exam paper. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday.